Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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4
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2answers
68 views

Last digit of $235!^{69}$

Problem What is the last digit of $235!^{69}$? It's been far too long since I did any modulo calcuations, and even then, the factorial would set me back. My initial thought goes to the last digit ...
3
votes
1answer
36 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
2
votes
2answers
47 views

Factorial Summation Definition

A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I ...
0
votes
2answers
38 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
0
votes
0answers
21 views

How to find the first and second number of factorial

How to find the first and second number of $40!$ Example $8!=40320$ the first is $4$ and second is $0$ I want to see the solution
5
votes
2answers
125 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
0
votes
2answers
69 views

The method of solving for a factor of $90!$ [duplicate]

If $90! = (90)(89)(88)...(2)(1)$, then what is the exponent of the highest power of $2$ which will divide $90!$ ? How would I apply one of the easiest method from Here? I need help on applying ...
13
votes
0answers
218 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
4
votes
5answers
152 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
2
votes
0answers
29 views

Factorial ratio sum of finite series

Given: $ S = \sum_{i=1}^{n-1}{i! \over n!} $ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $ S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275 $
92
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
1
vote
4answers
63 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
1
vote
2answers
34 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
5
votes
3answers
517 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
2
votes
4answers
78 views

Finding $\lim_{n \to \infty} \dfrac{n^n}{(2n)!}$

Struggling to apply Squeeze THM to find this limit. Specifically, I need a sequence which is always larger than the one in the problem, but which can easily be derived from the middle sequence.
0
votes
2answers
28 views

Factorial with names

Ok so, I have had an argument with my teacher over 1 quiz question that was marked wrong in my data management class. Question. Determine the number of ways that 12 members of the boys' baseball team ...
0
votes
1answer
45 views

Why is 0 factorial 1? [duplicate]

n factorial is product of all numbers between n and 1. 0 factorial is (0 * 1 = 0). Why is 0 factorial 1? How can I proof this in mathematical way?
0
votes
0answers
38 views

Limit of a sequence $a_n = \sqrt[n]{n!}/n$ [duplicate]

Find the limit of the sequence $$a_n = \frac{\sqrt[n]{n!}}{n}$$ I can figure out the limit of the sequence by letting $n=1,2,3,\dots$ but what would be the more conceptual approach to finding the ...
0
votes
0answers
16 views

Solving $(\prod^t_{i=1} N_i m_i)!$

I would like to know how to solve or simplify the factorial $(\prod^t_{i=1} N_i m_i)!$. Here, $i, N_i, m_i$ are positive integers. My effort: $$(\prod^t_{i=1} N_i m_i)!$$ $$\implies (\prod^t_{i=1} ...
3
votes
1answer
90 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
2
votes
2answers
54 views

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$

Prove $\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4 \cdots(2n)}<\frac{1}{\sqrt{2n+1}}$, $n\ge 1$. I begin by letting $n=1$ then $\frac{1}{2}<\frac{1}{\sqrt{3}}$. Then assume $\frac{1\cdot ...
0
votes
0answers
22 views

Find $\lim_{n \to \infty} \frac{2^n}{n!}$ - Stirling [duplicate]

Find $\lim_{n \to \infty} \frac{2^n}{n!}$. How is possible to solve this limite with Stirling formula? We can solve it with the ratio test, but I asked myself if it's possible with Stirling.
0
votes
1answer
23 views

Factorial grow faster than Exponential - permutation case

It is said that factorial grows faster than exponential, but in the case of permutation: ...
5
votes
1answer
68 views

Summation of factorials.

How do I go about summing this : $$\sum_{r=1}^{n}r\cdot (r+1)!$$ I know how to sum up $r\cdot r!$ But I am not able to do a similar thing with this.
0
votes
0answers
14 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 ...
2
votes
1answer
38 views

An inequality involving factorials and two variables

The problem is as follows: For $m\ge n>1$ prove that $$(m-2)!(n-1)+(n-2)!(m-1)+(m-2)(n-2)\ge (m-1)(n-1)$$ Since $(m-1)(n-1)-(m-2)(n-2)=m+n-3$ so we only need to show that ...
1
vote
3answers
58 views

How to isolate variable algebraically in a combinatorics equation?

How would I isolate a variable algebraically in a combinatorics equation? For example, if I'm given: $$C(k, 2) = 45$$ How would I solve for $k$, without trying random values of $k$? I know that ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
0
votes
2answers
93 views

Find all triples of non-negative Integers $a,b,c$ such that $a!b!=a!+b!+c!$ [duplicate]

Exactly what it says in the Title; not much development from there :/
1
vote
2answers
42 views

How to show that for all k, $k! \ge (k/2)^{k/2}$

I'm working on a homework problem that has me showing a "$\Omega(n\log k)$ lower bound on the number of comparisons needed to sort a sequence of $n$ elements, when the input sequence consists of ...
6
votes
1answer
76 views

When $n!=m(m+1)(m+2)$: A Diophantine Equation

I believe that I saw this problem not long ago in a book: Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers. However, I am no longer able to find this question, and ...
5
votes
0answers
39 views

Integer solutions for $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$

Consider the equation $\lceil{\sqrt{x!}}\rceil^2 - x! = y^2$. For $x \le 16$, the equation has the following integer solutions: $$ \begin{matrix} x = 0 & y = 0 \\ x = 1 & y = 0 \\ x = 4 ...
0
votes
1answer
22 views

Perform index shift on summation containing factorial

I'm having a hilariously hard time solving a problem that looks/feels so easy but just won't open up to me. I'm trying to show this equality: $$\sum_{k=0}^{n-1} \frac{1}{k!(n-k-1)!} = \sum_{k=1}^{n} ...
2
votes
3answers
56 views

Problem on factorials and divisiblity of number theory [closed]

How do I prove that $a!b!$ completely divides $(a+b)!$
8
votes
3answers
105 views

Last $500$ digits of $2015!-1$

As the title says, I'm looking for the last $500$ digits of $2015!-1$. I assume it's a repetition of zeroes from the factorial, so the final result is a lot of $9$-s, but I can't formulate a solution ...
0
votes
2answers
486 views

Powerball odds - factorial?

According to Powerball.com, the game is played like this ...we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls Their odds explain that the ...
2
votes
8answers
129 views

Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

The following series $$\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$$ converges. It fails the divergence test, but once I apply the ratio test, the limit is always equal to $\infty$. Unless you cannot ...
0
votes
1answer
41 views

identity with falling factorials

How can one show that $$\sum_{k=0}^n \frac{(n)_k}{k!} = 2^n$$ for all $n \geq 0$ where for $m \in \mathbb{Z}$ and $k \geq 0$ $(m)_k$ is the "falling factorial": $$(m)_k = \begin{cases} 1, ...
0
votes
1answer
44 views

Fraction Factorial [duplicate]

How do we find factorial of fractions? For eg: $\frac{1!}{2!}=(\frac{\pi}{4})^{\frac{1}{2}}$ Factorials are used in combinatorics and they can only be functioned on integers to give integers.Then how ...
0
votes
1answer
38 views

highest value of 'a'

I got a question when I started factorials Q. If $a^8$ and $8^a$ is completely divisible by $50!$ Then which one of the following is true about 'highest value of a'? (A) $10<a<14$ ...
2
votes
2answers
41 views

Factorial Representation of product

So I've been trying to work out if it is possible to write: $\large \Pi_{i=1}^n (3i-1)$ as an expression involving the quotient or product of two factorials, or really any expression involving ...
4
votes
0answers
42 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
2
votes
0answers
39 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
6
votes
4answers
109 views

Prove: $\forall$ $n\in \mathbb N, (2^n)!$ is divisible by $2^{(2^n)-1}$ and is not divisible by $2^{2^n}$

I assume induction must be used, but I'm having trouble thinking on how to use it when dealing with divisibility when there's no clear, useful way of factorizing the numbers.
0
votes
0answers
54 views

How many trailing zeroes does 4617! contain? [duplicate]

I am getting $1151$ as answer on continuous division by $5$. Is it right? On each division by 5, some remainder is generated...doesn't that count? Example: 4617/5 + 923/5 + 184/5 + 36/5 + 7/5 ...
0
votes
1answer
53 views

Can a factorial have an exponent

can you do $(4!)^4$ or do you have to do $4! \times 4$, I have looked on google but can't find anything to prove or disprove if it's possible.
7
votes
2answers
216 views

Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer

I need help to prove that the quantity of this division : $\dfrac{({10!})!}{{10!}^{9!}}$ is an integer number, using combinatorial proof
1
vote
0answers
29 views

Permutations to divide a solid

I have a 3-dimensional cuboid, with dimensions 2x2x1. I wish to divide this into smaller EQUAL sized cuboids of size 2x1x1. Then I want to extend this case for larger cuboids, assuming that equal ...
50
votes
11answers
34k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
33
votes
13answers
13k views

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is ...